I am really hoping this exists but I spent some time this afternoon trying to find it and couldn’t.
I was thinking mailgun works great when you want to send/receive email from your specific domain and can turn email control for that domain over to them completely. However, I was thinking it would be cool to have a service that will periodically check an IMAP box and then if a message is received, parse it into all of its parts and then send to a URL via JSON or even have a REST interface to retrieve them. I found context.io which is a free service doing this with data-mining by their company for their needs but was thinking something paid that would keep data private might be useful for any company dealing with inbound email in some form.